Shrinkwrap addon conflicting with curve bevel
I think you may be over complicating

Shrinkwrap modifier is used to project one flat surface over the another, not really very fit for complex animation, nor projecting non-sheet-like objects with volume.

Also curve objects are not really very fit for deformations (a least not in the Z axis), they will inevitably lead to artifacts due to their triangulated nature, lack of proper subdivisions, and non quad based topology.

You should use a base mesh object instead, which also widens the choice of modifiers available, and possibly use a Wave modifier instead, unless you need a very specific or precise wave shape.

Only then on top of the Wave deformation modifier (or whichever other you choose) apply a Solidify to add thickness and a Bevel to chamfer or fillet the corners.

That way you will guarantee that the thickness and bevels are not distorted as much by the Wave, nor squashed by the Shrinkwrap.

You may optionally add a Subdivision modifier set to Simple on top of the stack, so you get a smoother mesh with less bevel distortions